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- 1 The width of a slot of a duralumin forging

1. The width of a slot of a duralumin forging is ( in inches) normally
distributed with ¼ = 0.9000 and Æ = 0.0030. The specification limits
were given as 0.9000 ± 0.0050.
(a) What percentage of forgings will be defective?
(b) What is the maximum allowable value of Æ that will permit no more
than 1 in 100 defectives when the widths are normally distributed with
¼ = 0.9000 and?
2. One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that number 6 will appear between 150 and 200 times. If number 6 appears exactly 200 times, find the probability that number 5 will appear less than 150 times.
3. (a) Suppose that X is a normal random variable with mean 5.
If P(X > 9) = 0.2, find the variance of X.
(b) The average life of a bread-making machine is 7 years, with a standard
deviation of 1 year. Assuming that the lives of these machines
follow approximately a normal distribution , find
(i) the probability that the mean life of a random sample of 9 such
machines falls between 6.4 and 7.2 years
(ii) For what value to the right of which 15% of the means computed
from random samples of size 9 would fall.
4. In an experiment with a new tranquilizer, the pulse rates (per minute) of 12 patients were determined before they were given the tranquilizer and again 5 minutes later, and their pulse rates were found to be reduced on the average by 7.2 beats with a standard deviation of 1.8. At the level of significance 0.05, do we have significance evidence that
the mean pulse reduction with this tranquilizer is less than 9.0 beats? Also, find 95% confidence interval for the average pulse rate.
5. A medical research team conducted a study to test the effect of a cholesterol reducing medication. At the end of the study, the researchers found that of the 4700 subjects who took the medication, 301 died of heart disease. Of the 4300 subjects who took a placebo, 357 died of heart disease. At _ = 0.01, can you conclude that the death rate is lower for those who took the medication than for those who the placebo? Construct a 95% confidence interval for the two population proportions.
6. A manufacturer claims that the calling range (in miles) of its 900-MHz cordless telephone is greater than that of its leading competitor. You perform a study using 14 phones from the manufacturer had a mean 1275 with a standard deviation of 45. The mean for 16 similar phones from its competitor was 1250 with a standard deviation 30. At _ = 0.05, is there enough evidence to support the manufacturers claim? Assume the population variances are equal. Find a 95% confidence interval for the difference of two population means.
7. The random variable X has probability density function
f(X) = aX + bX2 for 0 < X < 1
0 elsewhere
If E(X) =0.6, then
(a) Find constants a and b
(b) Find P(X _ 0.5).
(c) Find V ar(X).
8. The joint density of X and Y is given by
f(X, Y ) = C(Y Ë X) e(ËY) for ËY < X < Y, Y>0
0 elsewhere
(a) Find C.
(b) Find marginal density functions for X and Y.
(c) Find E(X) and E(Y).
9. The joint density of X and Y is given by
f(X, Y ) = 6(1 Ë Y ) for 0 < X < Y < 1
0 elsewhere
(a) Find the covariance of X and Y.
(b) If U = X Ë Y, find the variance of U.
10. A sample study of the number of pieces of chalk the average professor uses was conducted at two universities. If 80 professors at one university averaged 12.5 pieces of chalk per month with a standard deviation of 2.4, while 60 professors at the other university averaged 11.3 pieces of chalk per month with a standard deviation of 3.3, test at the 0.05 level of significance whether the difference between the means is significant.
Find the p-value and use it to reach a conclusion regarding significance at the 0.05 level. Compute a 95% confidence interval for the difference of two population averages.

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